[Paper Review] An adaptive edge-based smoothed finite element method (ES-FEM) for phase-field modeling of fractures at large deformations

Fucheng Tian, Xiaoliang Tang+2 more|arXiv (Cornell University)|Sep 24, 2019

Numerical methods in engineering81 references62 citations

TL;DR

This paper proposes an adaptive edge-based smoothed finite element method (ES-FEM) coupled with the phase-field method (PFM) for simulating fracture in hyperelastic materials under large deformations. By integrating ES-FEM’s high accuracy and mesh distortion insensitivity with a multi-level adaptive mesh refinement strategy, the method achieves superior computational efficiency and robustness, successfully reproducing complex crack patterns including interface-induced crack deflection in rubber-like materials for the first time.

ABSTRACT

This work presents the Griffith-type phase-field formation at large deformation in the framework of adaptive edge-based smoothed finite element method (ES-FEM) for the first time. Therein the phase-field modeling of fractures has attracted widespread interest by virtue of its outstanding performance in dealing with complex cracks. The ES-FEM is an excellent member of the S-FEM family developed in combination with meshless ideas and finite element method (FEM), which is characterized by higher accuracy, softer stiffness, and insensitive to mesh distortion. Given that, the advantages of the phase-field method (PFM) and ES-FEM are fully combined by the approach proposed in this paper. With the costly computational overhead of PFM and ES-FEM in mind, a well-designed multi-level adaptive mesh strategy was developed, which considerably improved the computational efficiency. Furthermore, the detailed numerical implementation for the coupling of PFM and ES-FEM is outlined. Several representative numerical examples were recalculated based on the proposed method, and its effectiveness is verified by comparison with the results in experiments and literature. In particular, an experiment in which cracks deflected in rubber due to impinging on a weak interface was firstly reproduced.

Motivation & Objective

• To address the computational inefficiency and mesh distortion sensitivity in phase-field modeling of fractures under large deformations.
• To integrate the high-accuracy, distortion-insensitive ES-FEM with the phase-field method for improved simulation of complex crack evolution.
• To develop a multi-level adaptive mesh refinement strategy tailored for the coupling of ES-FEM and PFM, reducing computational cost.
• To validate the method on benchmark problems, including crack deflection at weak interfaces, which had not been previously simulated in this framework.
• To enable accurate, stable, and efficient simulation of fracture in hyperelastic materials such as rubber and hydrogels under large deformation.

Proposed method

• Adopted the Griffith-type phase-field model within the ES-FEM framework to model brittle fracture at large deformations.
• Employed the Neo-Hookean hyperelastic constitutive model to describe material behavior under large strain.
• Implemented a multi-level adaptive mesh refinement strategy based on the ha-PFM algorithm, dynamically adjusting element density around the crack zone.
• Utilized edge-based smoothing for strain computation, enhancing accuracy and reducing stiffness overestimation compared to standard FEM.
• Integrated the phase-field variable and displacement field via weak Galerkin formulation, solving the coupled system using the Newton-Raphson method.
• Reconstructed the strain tensor using spectral decomposition to prevent non-physical crack growth under compression.

Experimental results

Research questions

• RQ1 Can the phase-field method be effectively extended to large deformation problems using the ES-FEM framework?
• RQ2 How does the adaptive mesh refinement strategy improve computational efficiency without sacrificing accuracy in ES-FEM-PFM coupling?
• RQ3 Can the proposed method accurately simulate complex crack patterns such as crack deflection at weak interfaces in hyperelastic materials?
• RQ4 How does the ES-FEM-PFM coupling compare to standard FEM in terms of convergence and robustness under mesh distortion?
• RQ5 To what extent can the method reproduce experimental crack propagation behavior in rubber-like materials?

Key findings

• The proposed ES-FEM&APFM method successfully reproduced the experimental observation of crack deflection at a weak interface in a rubber-like material, a first-time simulation of its kind.
• The load-displacement curve from the simulation showed excellent agreement with experimental data and literature results, validating the method's accuracy.
• The method achieved approximately 20 times higher computational efficiency compared to non-adaptive ES-FEM due to the adaptive mesh refinement.
• The adaptive mesh refinement added only 1%–2% extra computational overhead while significantly improving accuracy and convergence.
• The method demonstrated superior robustness and convergence compared to standard FEM, which exhibited slow and unstable convergence under the same mesh conditions.
• The crack propagation patterns, including branching and deflection, closely matched those observed in experiments and prior simulations, especially in the panel with holes and interface-deflection cases.